Quantum mechanics takes the classical variables p and x that describe the state of a classical system into non-commuting operators that act on a wave function that describes the state of the quantum mechanical system.

For quantum field theory, the old QM wave function is now considered a quantum mechanical operator. This is a second quantization, hence the name. The old QM wave function is an operator that can create and destroy particles. It acts on a new type of state function tht includes the details about the particles.

Hi all,
The title said it all. My question is:
How is one to interpret the name second quantization ?
Which specifically is quantized twice ?
Best Wishes
Sincerely,
DaTario

Why the name Second Quantization? Because First Quantization was already taken. [A,B] is roughly the First Quantization. Second Quantization doesn't mean something is done twice (or else we would have called it Twice Quantized).

Well, we don't have "first quantization" and "second quantization". We just have QUANTIZATION, or "canonical formalism" if you prefer this terminology. Quantum mechanics is QUANTIZATION applied to classical dynamical systems with FINITELY MANY degrees of freedom, while Quantum field theory is QUANTIZATION applied to classical dynamical systems with INFINITELY MANY degrees of freedom, the latter a.k.a. "classical fields", or "finite dimensional nonunitary irreducible linear representations of the universal covering group of the restricted Poincaré group"...

I do not know who of you is right but shouldn't we keep the discussion onto a more intuitive level ? I mean, i assure you that 99,9% of the people (including science advisors and mentors) will not consider such an answer to be very clear. Ofcourse this does NOT mean that it is wrong.

I'm a bit confused by your reply. Are not the quantum fields one usually makes use of finite dimensional representations (vector, spinor, etc) of a non-compact group and hence non-unitary?

DaTario,

Part of the problem with the name second quantization is that it is something of a historical misnomer. Early quantum field theorists thought that they were somehow "quantizing" the wavefunction. For example, Dirac proposed his famous equation for the electron as a single particle relativistic wave equation. However, the relativistic quantum field describing the electron obeys the same equation. Confusion arose. I tend to think of it as something analogous to the way we still use the term "electromotive force" for something that isn't a force.

Ok, I will try and be a bit more clear at least on the infinite-dimensional representation part. To this end, I'm going to forget about untarity, covering spaces, Poincare etc., and I'm going to consider some toy examples.

Take ordinary physical space to be [tex]\mathbb{R}^3[/tex], and consider the action of rotations on fields defined in space.

An example of a scalar field defined on space is temperature [tex]T[/tex], which associates a temperature [tex]T \left( \vec{r} \right)[/tex] with every position [tex]\vec{r}[/tex] in space. In other words, [tex]T[/tex] is a function, with [tex]T: \mathbb{R}^3 \rightarrow \mathbb{R}[/tex], so [tex]T[/tex] is a member of the infinte-dimensional function space [tex]\{f: \mathbb{R}^3 \rightarrow \mathbb{R} \}[/tex].

Now let a rotation [tex]R[/tex] act on the space of scalar fields. Since [tex]R[/tex] operates (rotates) on (3-dim) vectors, this action is defined through a representative operator as follows. The representative of [tex]R[/tex] acts on a scalar field [tex]T[/tex] to give a new scalar field (i.e., another function) [tex]T'[/tex] such that the new temperature at the rotated positon [tex]\vec{r}' = R \vec{r}[/tex] is the same as the old temperature at the unrotated position [tex]\vec{r}[/tex]:

Now consider a field that assigns a vector [tex]\vec{E} \left( \vec{r} \right)[/tex] to each position [tex]\vec{r}[/tex] in space. [tex]\vec{E} \left( \vec{r} \right)[/tex] is an element of a 3-dimensional space (say) [tex]V[/tex] (to distinguish [tex]V[/tex] from the space of positions), but [tex]\vec{E}[/tex] itself is an element of the infinte-dimensional space [tex]\{ \vec{A}: \mathbb{R}^3 \rightarrow V \}[/tex] of vector-valued functions of position.

A rotation [tex]R[/tex] acts on the space as vector fields as follows. A rotation [tex]R[/tex] acts on the vector field [tex]\vec{E}[/tex] to give a new vector field [tex]\vec{E}'[/tex] such that the new field evaluated at the rotated position [tex]\vec{r}'[/tex] field is the same as the old field evaluated at the unrotated position and then rotated [tex]\vec{r}[/tex].
This makes sense because [tex]R[/tex] can act directly on [tex]\vec{E} \left(\vec{r} \right)[/tex], since [tex]\vec{E} \left(\vec{r} \right)[/tex] is an element of the 3-dimensional space [tex]V[/tex].

For both scalar and vector fields, finite-dimensional (1-dim for scalar fields, 3-dim for vector fields) have been used to get at the actual required infinite-dimensional representations.

It is an interesting mathematical exercise to show that [tex]R^{-1}[/tex], not [tex]R^[/tex], is need in the arguments in order to define a representation, i.e., a homomorphism of groups.

It is also interesting to go through this for the action of the Poincare group on "classical" Dirac spinor-valued fields on spacetime. I may try and give a pedagogical exposition of this, including unitarity, in another thread.

ps : i have been thinking to set up some kind of "general introduction to theoretical physics"-thread covering the intro of QFT all together with basic gauge symmetry and the implementation of group theory. I would like to invite you to participate (along with all others that like this idea) and check out some of the texts i already have written in my journal. What do you think ?

ps George, we can also cover the difference between QM and QFT , and "why we use fields" (see my journal). I have written a first attempt on this. I provided a link to that text in my first post of this thread. Let me know what you think of it, please. I would really appreciate it

George,
I'm a bit confused by your reply. Are not the quantum fields one usually makes use of finite dimensional representations (vector, spinor, etc) of a non-compact group and hence non-unitary?

I didn't notice your post until after I made my second post. My second post might shed some light, or it might just roil the waters! My examples are somewhat poor because I have used a compact group, but as I said in that post, I may start a thread and go through this (except for proving the irreducibility!) in some detail for "classical" Dirac fields

i have been thinking to set up some kind of "general introduction to theoretical physics"-thread covering the intro of QFT all together with basic gauge symmetry and the implementation of group theory. I would like to invite you to participate (along with all others that like this idea) and check out some of the texts i already have written in my journal. What do you think ?

Sounds very interesting.

I would like to participate, but I think and write *very* slowly, so I don't know how much this will happen.